Convergent scattering series solution of the inhomogeneous Helmholtz equation via renormalization group and homotopy continuation approaches

Abstract

The inhomogeneous Helmholtz equation can be represented by an equivalent integral equation of the Lippmann-Schwinger type. Since it is very costly to solve the Lippmann-Schwinger equation exactly via matrix inversion, researchers often try to use the popular Born series, which represents a physics based iterative solution. However, the popular Born series is only guaranteed to converge if the contrast volume is sufficiently small. We here use renormalization group (RG) and homotopy continuation (HC) approaches to derive novel scattering series solutions of the Lippmann-Schwinger equation which are guaranteed to converge for arbitrary large contrast volumes (perturbations). Our RG approach is based on the use of an auxillary set of scale-dependent scattering potentials and Green's operators which gradually evolve toward the real physical scattering potential and the physical Green's operator, respectively. We show that the auxillary Green's operators satisfies the renormalization group law and we derive the corresponding renormalization group flow equation by using familiar “scaling” arguments. The HC approach comes from pure mathematics and is based on the deformation of a relatively simple reference equation (initial condition) into a more complicated target equation (the Lippmann-Schwinger equation for the physical scattering potential), via a parameter that varies continuously between 0 and 1. Our HC approach is more general than the RG approach since it involves a convergence control operator, but the two approaches are nevertheless closely connected. We show that the RG flow and HC equations can be implemented very efficiently using the Fast Fourier Transform (FFT) algorithm. Also, we present a very efficient iterative scheme based on a combination of the zeroth-order deformation equation with a floating initial. We illustrate our results with a numerical example related with seismic wavefield modeling in strongly scattering media.